## 1. Connectives in Multi-valued Logic

There have been many generalisations of the classical two-valued logic to the multi-valued setting, since the seminal work of Lukasiewicz (1926).

My interests are more in the analytical and algebraic aspects of the connectives used in this logic, rather than the formal logics themselves. Specifically, I am interested in logics whose truth-values come from a lattice L and where the connectives could be expressed as truth-functionals. If the lattice L = [0,1], which is also a chain, then we obtain what is commonly termed as "Fuzzy Logic".

I am interested in the many connectives that arise naturally as generalisations of classical logic connectives. My current interests are in studying the following:

• Fuzzy Implications: Generation and Properties
• Functional Equations - specifically involving fuzzy logic connectives
• Triangular norms and their residuals
• Algebraic stuctures on families of Fuzzy Implications
• Fuzzy Implications (FIs) are hybrid monotonic binary operations on the unit interval [0,1]that are a generalization of the classical implication operator of the two-valued logic. FIs play asimilar role in inference much as their classical logic counterparts do and hence hold a centralplace among FLCs. The existing literature and studies on FIs from the above aspects have been covered by usin the research monograph Fuzzy Implications which is the first and only one this subject:

However, many interesting questions need to be answered, both on the theoretical and applicational sides. My endeavours are to pursue these questions.

## 2. Approximate Reasoning using Fuzzy Sets

Associated with fuzzy logic, is the theory of "Fuzzy Sets".

Approximate reasoning, as introduced by Zadeh in his early papers on fuzzy logic, has been paraphrased thus by Hellendoorn: "Approximate reasoning in its broadest sense is a collection of techniques for dealing with inference under uncertainty in which the underlying logic is approximate or probabilistic rather than exact or deterministic."

An Inference mechanism in approximate reasoning can be seen as a function which derives a meaningful output from imprecise inputs. Approximate reasoning schemes involving fuzzy sets are one of the best known applications of fuzzy logic in the wider sense. Fuzzy Logic connectives play an important role in these schemes.

My current research interests are in studying the following:

• Suitability of employing different fuzzy logic connectives in the various fuzzy inference mechanisms, especially Fuzzy Relational and Similarity Based Reasoning schemes.
• Theoretical models that are computationally efficient without loss of approximation capability.

## 3. Issues in High Dimensional Data Analysis

In recent times, analysis of High Dimensional (HD) data has become a major area of study. Some effects due to increasing dimensionality have been known for some time now, for instance the combinatorial explosion in search space. However, studies done during the last decade have revealed many new aspects of HD data and their impact on various algorithms and processes in Data Analysis.

One such aspect is the Concentration of Norms phenomenon, which is the inability of distance measures to separate points well in high dimensions because of which the distances between any pair of points appear more or less equal. Another is the hubness phenomenon, wherein some data points appear to be more preferred than others in high dimensions.

While the existence of the above problems has been proven both emperically and theoretically, only nascent efforts have been made towards both analysing them and finding solutions.

My current research interests are in investigating these problems and along two approaches.

• How can one mitigate these effects in practical algorithms, like clustering of high dimensional data ?
• A theoretical analysis and proposal of potential solutions that could at least be provably useful in certain settings.